A151577 -id:A151577 - cp's OEIS frontend

A334218 Triangle read by rows: T(n,k) is the number of permutations of 1..n arranged in a circle with exactly k descents.

1, 1, 0, 0, 2, 0, 0, 3, 3, 0, 0, 4, 16, 4, 0, 0, 5, 55, 55, 5, 0, 0, 6, 156, 396, 156, 6, 0, 0, 7, 399, 2114, 2114, 399, 7, 0, 0, 8, 960, 9528, 19328, 9528, 960, 8, 0, 0, 9, 2223, 38637, 140571, 140571, 38637, 2223, 9, 0, 0, 10, 5020, 146080, 882340, 1561900, 882340, 146080, 5020, 10, 0

Offset: 0

Andrew Howroyd, May 04 2020

			Triangle begins:
  1;
  1, 0;
  0, 2,   0;
  0, 3,   3,    0;
  0, 4,  16,    4,     0;
  0, 5,  55,   55,     5,    0;
  0, 6, 156,  396,   156,    6,   0;
  0, 7, 399, 2114,  2114,  399,   7, 0;
  0, 8, 960, 9528, 19328, 9528, 960, 8, 0;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Columns k=2..9 are A027540(n-1), A151576, A151577, A151578, A151579, A151580, A151581, A151582.
Row sums are A000142.
Cf. A008292.

T(n, k) = {if(n==0, k==0, n*sum(j=0, k, (-1)^j * (k-j)^(n-1) * binomial(n, j)))}

T(n, k) = n*A008292(n-1, k) for n > 1.
T(n, k) = T(n, n-k) for n > 1.
T(n, k) = n*Sum_{j=0..k} (-1)^j * (k-j)^(n-1) * binomial(n, j) for n > 0.

A151576 Number of permutations of 1..n arranged in a circle with exactly 3 adjacent element pairs in decreasing order.

Original entry on oeis.org

0, 4, 55, 396, 2114, 9528, 38637, 146080, 526240, 1831644, 6217523, 20716164, 68059710, 221195824, 712856665, 2282058360, 7266358556, 23035517940, 72760054815, 229112753980, 719545590010, 2254604460264, 7050252659525, 22006821057936, 68581455012504, 213411502891468

Offset: 3

R. H. Hardin, May 21 2009

Exactly 2 adjacent element pairs in decreasing order gives A027540(n-1).

Andrew Howroyd, Table of n, a(n) for n = 3..500
Index entries for linear recurrences with constant coefficients, signature (16,-111,438,-1083,1740,-1817,1190,-444,72).

Column k=3 of A334218.
Related sequences: A151577-A151610.
Cf. A000460.

a(n)={n*(3^(n-1) - n*2^(n-1) + n*(n-1)/2)} \\ Andrew Howroyd, May 05 2020

From Andrew Howroyd, May 05 2020: (Start)
a(n) = n*A000460(n-1).
a(n) = n*(3^(n-1) - n*2^(n-1) + n*(n-1)/2).
a(n) = 16*a(n-1) - 111*a(n-2) + 438*a(n-3) - 1083*a(n-4) + 1740*a(n-5) - 1817*a(n-6) + 1190*a(n-7) - 444*a(n-8) + 72*a(n-9).
G.f.: x^4*(4 - 9*x - 40*x^2 + 131*x^3 - 98*x^4)/((1 - x)^4*(1 - 2*x)^3*(1 - 3*x)^2).
(End)

Terms a(18) and beyond from Andrew Howroyd, May 05 2020

cp's OEIS Frontend

A334218 Triangle read by rows: T(n,k) is the number of permutations of 1..n arranged in a circle with exactly k descents.

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